There's only a handful of exceptions to this rule:
(1) Cape Sagres (off Conakry, Guinea) Because of the date (1461) some people think it is in honor of Prince Henry. But it wasn't. It was named because the captain Pedro de Sintra had no translators to acquire the native name, and he had ran out of geographical names. (He wanted to call it "red cape", but he had already used vermelho & roxo for previous red capes on that journey, and so resorted to "Sagres" because it looked like Cape Sagres in Portugal. The only instance of an overseas toponym named after a location in Portugal)
(2) Rio do Infante - (Great Fish River, RSA), named after Prince Afonso. The only genuine vanity name after a Portuguese royal, imposed by king's order on maps after Afonso's tragic death in 1491. Very exceptional.
(3) Principe Island - named after same prince Afonso (daddy was distraught); but it was not an original honor but to correct a mistake, a re-naming after the original island's name, Santo Antao, caused confusion with Santo Antao in Cape Verde. Forced necessity.
(4) Fort Manuel, in Cochin - another forced necessity to correct a mistake; re-naming after the original fort's name, Sant Iago, caused confusion with Sant'Iago in Kilwa. This is the ONLY locality named after a king in the Portuguese empire that I have found.
(5) Vila da Rainha in the captaincy of Espirito Santo. Can't quite figure that one out, perhaps she was the financier?
(6) "Nova Lusitania" - the only instance of a "New X" in the Portuguese empire. It didn't stick. It was the name officially chosen by the donatary captain Duarte Coelho to his captaincy of Pernambuco. It didn't work. Despite his efforts, everyone, colonists and sailors alike, still called it Pernambuco, and it was immediately officially renamed "Pernambuco" after Coelho died. Never used again.
Now, you may what about "Nova Lisboa" in Ceara and "Nova Lisboa" in Angola? Well, those came only after the 1580 Hapsburg union when Spanish naming habits started influencing Portuguese toponyms. But note that the Ceara settlement was promptly renamed "Fortaleza" once they were gone.
So, other than these few exceptions (out of hundreds upon hundreds), Portuguese overseas toponyms were never named after places in Portugal or rulers in Portugal.
SPANISH
It is the Spaniards who inaugurated the tradition of "New X" and royal names.
Columbus was thoroughly unsystematic in his nomenclature. Columbus named a lot of places by a variety of criteria, e.g. saintly-liturgical names loom large, like San Salvador, San Cristobal, San Juan Bautista, Dominica, etc. and vanity name (e.g. for ship Mariagalante). But he more rarely used physical names and practically no local names at all (Jamaica & Cuba can be cited as exceptions, but he didn't want these for them).
More importantly, Columbus was also the first to use impose toponyms that corresponded to location names in Spain - Hispaniola, Guadalupe, Monsterrat, etc. And also the first to impose royal names - Fernandina, La Iabella, Juana.
(Side Note: another point against those weird people who imagine Columbus was Portuguese or a decent cartographer for that matter - he didn't follow the traditional Portuguese naming pattern. Those fringe folks who continue to base their claims on the ludicrous notion that Cuba is named after the village of Cuba in Portugal should take note of how completely unprecedented and completely against all Portuguese tradition that would have been. If anything, it is proof positive he wasn't Portuguese.)
The invocation of Spanish localities would multiply and permeate a lot of Spanish toponyms. e.g. Castilla de Oro, Cartagena das Indias, Nueva Andalusia, Nueva Granada, Nueva Cordoba, Nueva Espana, Nueva Castilla, Nueva Leon, Nueva Salamanca, Nueva Galicia, Nueva Toledo, etc. And royals continue too, e.g. Philippines.
ENGLISH
The English took the Spanish style and ran with it to extremes.
If Spanish loved "Nueva", English just adored "New". This, of course, is most blatant in the New World. So we have New Plymouth, New England, New Albion, New Hampshire, New Bedford, New York, New Jersey, New Haven, New Cambriol, Nova Scotia, New London, Newtown, Newport, New South Wales, etc. Everything new. And when not new, named after English localties regardless - Boston, Salem, Dorchester, Weymouth, Roxbury, Waterford, Lynn, Portsmouth, Rye, Dover, Gloucester, Ipswich, Cambridge, Sudbury, Newbury, Hartford, Greenwich, Thames River, Ipswich River, Isle of Wight, etc. or some other Old World places (Rhode Island, Avalon, Maine, Alexandria, New Brunswick, etc.)
And if Spaniards liked to their kings, the English couldn't get enough of honoring them. Virginia, Jamestown, Maryland, Carolina, half a dozen Charlestons, Charles River, Georgia, etc. And they didn't forget to honor their lords back home with their Falkland, Cromwell, Melbourne, Richmond, Delaware, Hampton, Salisbury, Sydney, etc.
I suppose a lot of this was inevitable by the Protestant lack of saints. Had to draw from another barrel.
But what is starkly missing here are native names. Now, a few places started off with native names (e.g. Naumkeag, Saugus, Wessagusset) but they didn't last long and were quickly given proper English names. I guess the traditional stereotype of English colonialism as distancing, racialist separation started early.
Of course, doesn't exactly apply to the East Indies. Although the English established forts in India with Anglo names (Fort St. George, Fort William, Fort St. David), the underlying locations had well-established local names and generally overwhelmed them (Madras, Calcutta, Cuddalore). But the choices of the West Injuns were largely ignorable.
Of the few native names in America that lasted in English dominions - e.g. Massachusetts, Connecticut, Nantucket, Narangasset, - most were actually not bestowed by the English, but rather by the Dutch, on their prior mapping expeditions, and were already on the charts. Siimilar to the more westerly indian names - Niagara, Ottawa, Michigan, Illinois, Chicago, Peoria, Ohio, Arkansas, Iowa, Mississippi, Missouri, Wisconsin, etc. - thank the New France explorers for putting those names down first, else they would likely have gone the way of anglo western names like Pittsburgh, Cincinnati, Cleveland.
SWEDES
Swedes, when they colonized, were as bad as the English, if not worse. All Nya this and Nya that - Nya Sverige, Nya Stockholm, Nya Alfsborg, Nya Goteborg, Nya Korsholm, Nya Vasa, Sveaborg, Torne, Finland, Uppland, Forts Cristina, Trefaldighets, etc. They didn't maintain any native names anywhere.
DUTCH
The Dutch had "Nieuws" and back-home localities - New Netherlands, New Amsterdam, New Holland, New Amersfoort, Middelburg, Flushing, Vrissendael, Harlem, Zeelandia - and liked to repetitively honor their rulers back home (Oranges, Nassau, Wilhelm, Mauritius, Casimir, van Verre, Staten) and they didn't seem to care about repetition (gotta be some half-dozen Fort Oranges and Nassaus apiece). Not too many local captain vanity names (some exceptions, e.g. Rensselaerwyck, Bronx, Pavonia)
But the Dutch were much more adept at maintaining local names. Not, however, local names of localities as used by the indians, but rather names of the indian tribes themselves. Maspeth, Canarsee, Hackensack, Hoboken, Passaic, Montauk, Shinnecock, Paugussett, Quinnipac, etc. So it corresponds more closely to Portuguese criterion (2) rather than (1). They also were prone to occasional physical names (Noten Is., Brooklyn, Sandy Hook, etc.), but apparently not as first preference.
FRENCH
Haven't looked yet at that too closely. But from the top of my head, seems they followed a semi-Portuguese/semi-Dutch pattern at first. Initially, heavily into native names, like Hochelaga, Stadacona, Tadoussac, Saguenay, Quebec, Canada, Ottawa, Illinois, Miami, Niagara, Ticonderoga, Penobscot with the occasional but not as systematic saints (e.g. St. Croix, St. Lawrence, Sault St. Marie, St. Louis (repeated several times), St. Joseph, St. Jean, St. Anne). But, like the Dutch, occasionally used home-invoking names (e.g. Cape Breton, Terre des Bretons, Saint-Malo, Place Royal, Brion).
Later adopted some Spanish-style honor mode, with New France, France Antartique, France Equinoxiale, New Angouleme, New Orleans, Louisiana, Louisville, Louisbourg, Henriville, Richelieu, Coligny, Forts Bourbon, Dauphin, La Reine, Chartres, Toulouse, etc.
Portuguese-style vanity names (i.e. after local captains, rather than honoring someone back home) seemed to come and go. Several old native/saintly-named localities were later renamed after captains (e.g. Cataracoui to Frontenac, St. Louis to Chambly, Ticonderoga to Carillon?). Vanity seemed a strong factor in the names along the De La Salle expeditition of the 1670s. The 1740s Allengheny rush seems to have also been heavily personal, with Le Bouef, Machault & Duquesne, all in a row. But perhaps this should be more carefully looked at to differentiate between vanity names (local captains) & honor names (lords back in France).